Sunday, 13 May 2012

Something from nothing: Competing premises

In my previous post, I discussed how Dr. Carrier's argument that the universe could come into existence from nothing is cogent only if you choose to assume that anything can happen in preference to assuming that there is no dimension of time in a state of absolute nothingness.

My own viewpoint is that absolute nothingness does in fact imply a lack of a time dimension, and so Carrier's argument does not hold. So how can we choose which is the better description of absolute nothingness? In this post, I will argue for my position.

To a large extent, it's a matter of taste. I can't prove that my interpretation of absolute nothingess is any more correct than Carrier's. However I do believe it is the more natural interpretation. Carrier's nothingness is an explosion of wild creativity, creating everything that could possibly exist, while mine is static and unchanging. The latter seems more like nothingness to me at any rate.

Furthermore, I would add the point that Carrier's nothingness seems to be ill-defined. This is illustrated first by the fact that we can both infer such wildly different conclusions from the initial premise. But there are other problems with Carrier's argument.

Carrier is interested in the possibility that universes will be created, so he only considers the probability of the creation of universes. He assumes that x universes will be created, and therefore assumes that all we need to do is calculate the likely value of x given that all outcomes are equally likely.

However, making such calculations is subtly dependent on how you categorise the outcomes. If I described the possibilities as "Either a law prohibiting universes comes into existence or it doesn't", then it appears that there is a 50% chance that there will be no universes at all.

50% isn't so bad though, right? If my argument is correct, it's still quite likely that we would exist. After all, the alternatives appear to be zero universes or infinite universes, the latter guaranteeing our existence.

However, I can do the same thing that Carrier has done and break this rule up into an infinite number of alternatives, each implying that no universe exists. To substitute my idea into Carrier's argument:

Of all the logically possible things that can happen when nothing exists to prevent them from happening, continuing to be nothing is one thing, a law preventing universes of less than one dimension popping into existence is another thing, a law preventing universes of less than two dimensions popping into existence is yet another thing, and so on all the way to laws preventing universes of less than infinite dimensions, and likewise for every cardinality of infinity, and every configuration of dimensions.

Following the pattern Carrier's logic, we would then be forced to conclude that it is impossible for any universe to exist, as it is infinitesimally improbable that there would not be a law to prevent the existence of a universe with less than an infinite number of dimensions.

This argument is obviously facetious and incorrect. But, in my opinion, no more so than Carrier's. The reason is because the mathematical problem Carrier is attempting to solve is not well-defined. When there are no rules, you can conclude anything you like, depending on how you partition the various possibilities you assume are equally likely.

And this assumption, stated in Carrier's P2, is also problematic. "If there was absolutely nothing, then…nothing existed to prevent anything from happening or to make any one thing happening more likely than any other thing." Carrier develops this to mean that all outcomes are equally likely, however I do not believe this is justified.

The simplest way to explain Carrier's justification of this is to quote him when he expresses his statement in mathematical terms.

I believe here x is the probability of one outcome, y is the probability of another. I think he probably means to suggest "not greater than" by his usage of "/>" but this is a novel notation to me.

His reasoning appears to be flawless. If x is not greater than y, and y is not greater than x, then x equals y.

Well, no. Actually there is one situation where that is not the case, and that is when x and y are undefined. If they are undefined, you are not entitled to make any meaningful statement about them. 5/0 is not greater than 4/0. 4/0 is not greater than 5/0. 4/0 is not equal to 5/0. Both are undefined. Programmers will be familiar with these rules if they have ever encountered the special value NaN (not a number).

And I believe this is the case for Carrier's probabilities. I have shown that by simply assuming that any outcome is equally possible, one can draw contradictory conclusions by simply changing how you enumerate those possibilities. I feel this is probably consistent with probabilities being undefined rather than equal.

Carrier's P2 can probably be used to prove that any desired outcome is true, and so I suspect it is not well-defined mathematically or logically. Since the rules of logic must exist, Carrier's P2 cannot be true until it is defined rigorously enough to avoid these criticism. This is why I feel it is not quite as reasonable as my P2 which states that nothing happens because there is no time dimension. This latter P2 is obviously embarrassingly trivial to describe and reason about.